# Symmetric group on finite set has Sylow subgroup of prime order

From Groupprops

## Statement

Suppose is a natural number greater than . Let be the Symmetric group (?) of degree . Then, there exists a prime such that has a -Sylow subgroup (?) of order , i.e., it is a Group of prime order (?).

## Facts used

- Bertrand's postulate: The version we use states that for any , there exists a prime such that .
- Sylow subgroups exist

## Proof

The symmetric group of degree has order . By fact (1), there exists a prime such that . Thus, the largest power of dividing is . In particular, this means that the order of the -Sylow subgroup (which exists by fact (2)) is .